Confidence Intervals for One Population Mean (σ Known)

Confidence Intervals for One Population Mean (σ Known)

Point Estimation of the Population Mean

When estimating the mean (μ) of a population, the sample mean (\( \bar{x} \)) is used as a point estimate. A point estimate is a single value that serves as an estimate of an unknown population parameter.

• Point Estimate: The value of the sample mean, \( \bar{x} \), is used to estimate the population mean, μ.

• Sampling Error: The difference between the sample mean and the true population mean, expressed as \( \bar{x} - \mu \).

• Example: If a random sample of 200 adult American men has a mean height of 69.3 inches, then 69.3 is the point estimate for μ.

Margin of Error and Confidence Level

The margin of error (E) quantifies the maximum expected difference between the sample mean and the true population mean, given a specified level of confidence. The confidence level (commonly 90%, 95%, or 99%) indicates the probability that the interval estimate will contain the true population mean.

• Margin of Error (E): The maximum likely difference between \( \bar{x} \) and μ for a given confidence level.

• Confidence Level: The probability (as a percentage) that the confidence interval contains the true population mean. For example, a 95% confidence level means that 95% of similarly constructed intervals from repeated samples would contain μ.

• Alpha (\( \alpha \)): The probability that the interval does not contain μ. Calculated as \( 1 - \text{confidence level} \). For 95% confidence, \( \alpha = 0.05 \).

• Two-Tailed Probability: Since the sample mean can be unusually high or low, \( \alpha \) is split equally into both tails: \( \alpha/2 \) in each tail.

Diagram Explanation

The confidence interval is centered at the unknown population mean μ, with equal probability in both tails:

• Probability in each tail: \( P = \alpha/2 \) (e.g., 0.025 for 95% confidence)

• Central probability: \( P = 1 - \alpha = 0.95 \)

• Interval: \( \mu - E \leq \bar{x} \leq \mu + E \)

Formula for Margin of Error (σ Known)

When the population standard deviation (σ) is known, the margin of error for estimating μ is calculated as:

$E = z_{\alpha/2} \frac{\sigma}{\sqrt{n}}$

• \( z_{\alpha/2} \): The critical value from the standard normal (z) distribution corresponding to the desired confidence level (e.g., 1.96 for 95% confidence).

• \( \sigma \): The population standard deviation (assumed known).

• \( n \): The sample size.

Example Calculation:

• Given: \( \sigma = 2.8 \), \( n = 200 \), \( z_{0.025} = 1.96 \)

• Compute margin of error:

$E = 1.96 \times \frac{2.8}{\sqrt{200}} \approx 1.96 \times 0.198 = 0.39$

Constructing the Confidence Interval

The confidence interval for the population mean is constructed by adding and subtracting the margin of error from the sample mean:

• Interval Notation: \( \bar{x} \pm E \)

• Endpoint Notation: \( (\bar{x} - E, \bar{x} + E) \)

• Inequality Notation: \( \bar{x} - E < \mu < \bar{x} + E \)

Example: With \( \bar{x} = 69.3 \) and \( E = 0.39 \):

• \( 69.3 \pm 0.39 = (68.91, 69.69) \)

• Interpretation: We are 95% confident that the true mean height of adult American men is between 68.91 and 69.69 inches.

Using Technology (Calculator)

Most scientific calculators and statistical software can compute confidence intervals directly. For example, using the ZInterval function:

• Select Stats input mode.

• Enter: population standard deviation (σ), sample mean (\( \bar{x} \)), sample size (n), and confidence level.

• The calculator outputs the confidence interval endpoints.

• To find the margin of error, subtract the sample mean from the upper endpoint.

Summary Table: Steps for Constructing a Confidence Interval for μ (σ Known)

Step	Description	Formula/Action
1	Choose confidence level (e.g., 95%)	Set \( \alpha = 1 - \text{confidence level} \)
2	Find critical value \( z_{\alpha/2} \)	Use z-table or calculator (e.g., 1.96 for 95%)
3	Compute margin of error (E)	$E = z_{\alpha/2} \frac{\sigma}{\sqrt{n}}$
4	Construct confidence interval	\( \bar{x} \pm E \) or \( (\bar{x} - E, \bar{x} + E) \)

Important Notes

• The method above assumes the population standard deviation (σ) is known. In practice, σ is rarely known; when it is unknown, the t-distribution is used instead (see next section in your course).

• The confidence interval provides a range of plausible values for μ, not a probability that μ is in the interval for a specific sample.

• The width of the confidence interval depends on the confidence level, sample size, and population standard deviation.

Additional info: In real-world applications, σ is often estimated from the sample, leading to the use of the t-distribution for constructing confidence intervals. The above method is foundational for understanding more advanced estimation techniques.